Hereditary rings containing an injective maximal left ideal. (English) Zbl 0803.16010

R. Yue Chi Ming has proposed the following questions in his earlier papers. (i) If a ring \(R\) contains an injective maximal left ideal and if every maximal left ideal of \(R\) is projective, then is \(R\) semisimple Artinian? (ii) If \(R\) is a left hereditary ring containing an injective maximal left ideal, is \(R\) semisimple Artinian?
The authors answer both the questions in the negative by constructing a hereditary ring \(R\) containing an injective maximal left ideal as well as an injective maximal right ideal such that \(R\) is Artinian with nonzero Jacobson radical \(J\) with \(J^ 2 = 0\). They also give two new characterizations of semisimple Artinian rings in terms of left hereditary rings containing an injective maximal left ideal.


16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
16D25 Ideals in associative algebras
16P20 Artinian rings and modules (associative rings and algebras)
Full Text: DOI


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